This thesis studies the behavior of particles that are maximal at time t in
branching random walk and branching Brownian motion on R, for large values of
t. Precisely, we look at the behavior of the maximum in a branching random walk
in a time-inhomogeneous environment, where the law of the increments varies
with respect to time. We compare with known or simplified models such as the
model where random walks are taken to be i.i.d. and the branching random walk
in a time-homogeneous environment model. We then take a look at the correlations
between maximal particles in a branching brownian motion. Specifically, we
look at the branching time between those maximal particles. Finally, we apply
results and methods from the first chapters to study those same correlations in
branching Brownian motion in a inhomogeneous environment. The thesis’ main
result establishes existence of branching time at the center of the interval [0, t] for
the branching Brownian motion in a inhomogeneous environment, which is not
the case for standard branching brownian motion.We also present results of simulations
that agree with theoretical results and help establishing new hypotheses
for future research.